Gyroelongated square cupola explained

Type:	Johnson J22 - J23 - J24
Faces:	3x4+8 triangles 1+4 squares 1 octagon
Edges:	44
Vertices:	20
Symmetry:	C4v
Vertex Config:	4(3.4^3) 2.4(3^3.8) 8(3^4.4)
Dual:	-
Properties:	convex
Net:	Johnson solid 23 net.png

In geometry, the gyroelongated square cupola is one of the Johnson solids (J₂₃). As the name suggests, it can be constructed by gyroelongating a square cupola (J₄) by attaching an octagonal antiprism to its base. It can also be seen as a gyroelongated square bicupola (J₄₅) with one square bicupola removed.

Area and Volume

The surface area is,

A = \left( 7+2 \sqrt{ 2 } +5 \sqrt{ 3 }\right)a ^{2 ≈} 18 . 4886811 ...a ^{2.}

The volume is the sum of the volume of a square cupola and the volume of an octagonal prism,

V = \left( 1+ (2)/(3) \sqrt{ 2 } + (2)/(3) \sqrt{ 4+2 \sqrt{ 2 } +2 \sqrt{ 146+103 \sqrt{ 2 }}}\right)a ^{3 ≈ 6.2107658...a} ^{3.}

Dual polyhedron

The dual of the gyroelongated square cupola has 20 faces: 8 kites, 4 rhombi, and 8 pentagons.